Final answer:
After 15 years, which is equivalent to three half-lives of an isotope with a half-life of 5 years, only 12.5% or 1/8 of the original sample remains. This fraction is determined by the principle that after each half-life, the amount of isotope remaining is reduced by half.
Step-by-step explanation:
If the half-life of a certain isotope is 5 years, after 15 years, 3 half-lives would have passed (15 years ÷ 5 years per half-life). To calculate what fraction of the isotope would remain after 15 years, we use the principle that after each half-life, half of the isotope remains.
Therefore, after one half-life (5 years), 50% remains; after two half-lives (10 years), 25% (which is half of 50%) remains; after three half-lives (15 years), we would have half of 25%, which is 12.5%. The fraction of the isotope that will remain after 15 years is thus 0.125 or ⅛.
This decay process is exponential, not linear, meaning that each half-life reduces the remaining amount by half, regardless of the total time passed.